3. BASIC PROCESSES

A bound electron in an ion can be brought into a higher, excited energy
level through a collision with a free electron or by absorption of a photon.
The latter will be discussed in more detail in
Sect. 6.
Here we focus upon excitation by electrons.

The cross section Qij for excitation from level
i to level j for this
process can be conveniently parametrised by

(1)

where U = Eij / E with
Eij the excitation energy from level i to
j, E the energy of the exciting electron,
EH the Rydberg
energy (13.6 eV), a0 the Bohr radius and
wi the statistical weight
of the lower level i. The dimensionless quantity
(U) is the so-called
collision strength. For a given transition on an iso-electronic sequence,
(U) is not a
strong function of the atomic number
Z, or may be even almost independent of Z.

Mewe (1972)
introduced a convenient formula that can be used to
describe most collision strengths, written here as follows:

(2)

where A, B, C, D and F are parameters
that differ for each transition. The expression can be integrated
analytically over a Maxwellian electron distribution, and the result can
be expressed in terms of exponential integrals. We only mention here
that the total excitation rate Sij (in units of
m-3 s-1) is given by

(3)

with yEij / kT and
(y) is the
Maxwellian-averaged collision strength. For low temperatures, y
>> 1 and
(y) =
A + B + C + 2D, leading
to Sij ~ T-1/2e-y. The excitation rate drops exponentially due to
the lack of electrons with sufficient
energy. In the other limit of high temperature, y << 1 and
(y) =
-F lny and hence
Sij ~ T-1/2 lny.

Not all transitions have this asymptotic behaviour, however. For instance,
so-called forbidden transitions have F = 0 and hence have much
lower excitation rates at high energy. So-called spin-forbidden
transitions even have A = B = F = 0.

In most cases, the excited state is stable and the ion will decay back
to the ground level by means of a radiative transition, either directly
or through one or more steps via intermediate energy levels. Only in
cases of high density or high radiation fields, collisional excitation
or further radiative excitation to a higher level may become important,
but for typical cluster and ISM conditions these processes are not
important in most cases. Therefore, the excitation rate immediately
gives the total emission line power.

Collisional ionisation occurs when during the interaction of a free
electron with an atom or ion the free electron transfers a part of its
energy to one of the bound electrons, which is then able to escape from
the ion. A necessary condition is that the kinetic energy E of
the free electron must be larger than the binding energy I of the
atomic shell from which the bound electron escapes. A formula that gives
a correct order of magnitude estimate of the cross section
of
this process and that has the proper asymptotic behaviour (first
calculated by Bethe and Born) is the formula of
Lotz (1968):

(4)

where ns is the number of electrons in the shell and
the normalisation a = 4.5 × 10-24 m2
keV2. This equation shows that high-energy
electrons have less ionising power than low-energy electrons. Also, the
cross section at the threshold E = I is zero.

The above cross section has to be averaged over the electron distribution
(Maxwellian for a thermal plasma). For simple formulae for the cross section
such as (4) the integration can be done analytically and the result
can be expressed in terms of exponential integrals. We give here only the
asymptotic results for CDI, the total number of direct
ionisations per unit volume per unit time:

(5)

and

(6)

For low temperatures, the ionisation rate therefore goes exponentially
to zero. This can be understood simply, because for low temperatures
only the electrons from the exponential tail of the Maxwell distribution
have sufficient energy to ionise the atom or ion. For higher
temperatures the ionisation rate also approaches zero, but this time
because the cross section at high energies is small.

For each ion the direct ionisation rate per atomic shell can now be
determined. The total rate follows immediately by adding the
contributions from the different shells. Usually only the outermost two
or three shells are important. That is because of the scaling with
I-2 and I-1 in (5) and (6),
respectively.

This process is very similar to collisional ionisation. The difference
is that in the case of photoionisation a photon instead of an electron
is causing the ionisation. Further, the effective cross section differs
from that for collisional ionisation. As an example
Fig. 2 shows the cross section for neutral iron
and Na-like iron. For the more highly ionised iron the so-called
"edges" (corresponding to the ionisation potentials
I) occur at higher energies than for neutral iron. Contrary to
the case of collisional ionisation, the cross section at threshold for
photoionisation is not zero. The effective cross section just above the
edges sometimes changes rapidly (the so-called Cooper minima and
maxima).

Figure 2. Photoionisation cross
section in barn
(10-28 m-2) for Fe I (left) and Fe XVI (right).
The p and d states (dashed and dotted) have two lines each because of
splitting into two sublevels with different j (see
Sect. 2.1).

Contrary to collisional ionisation, all the inner shells now have the
largest cross section. For the K-shell one can approximate for
E > I

(7)

For a given ionising spectrum F(E) (photons per unit
volume per unit energy) the total number of photoionisations follows as

(8)

For hydrogenlike ions one can write:

(9)

where n is the principal quantum number,
the fine structure constant
and a0 the Bohr radius. The Gaunt factor
g(E, n) is of order unity and
varies only slowly as a function of E. It has been calculated and
tabulated by
Karzas &
Latter (1961).
The above equation is also applicable to
excited states of the atom, and is a good approximation for all excited
atoms or ions where n is larger than the corresponding value for
the valence electron.

Scattering of a photon on an electron generally leads to energy transfer
from one of the particles to the other. In most cases only scattering on
free electrons is considered. But Compton scattering also can occur on
bound electrons. If the energy transfer from the photon to the electron
is large enough, the ionisation potential can be overcome leading to
ionisation. This is the Compton ionisation process.

In the Thomson limit the differential cross section for Compton
scattering is given by

(10)

with the scattering angle and
T the Thomson
cross section (6.65 × 10-29 m-2). The energy
transfer E
during the scattering is given by (E is the photon energy):

(11)

Only those scatterings where
E >
I contribute to the ionisation. This
defines a critical angle c, given by:

(12)

For E >> I we have
(E)
T
(all scatterings
lead in that case to ionisation) and further for
c we have
(E)
0. Because for most ions I <<
mec2, this last condition occurs
for E
(Imec2 / 2)1/2 >>
I. See Fig. 3 for an example of some cross
sections. In general, Compton ionisation is important if the ionising
spectrum contains a significant hard X-ray contribution, for which the
Compton cross section is larger than the steeply falling photoionisation
cross section.

As we showed above, interaction of a photon or free electron with an
atom or ion
may lead to ionisation. In particular when an electron from one of the inner
shells is removed, the resulting ion has a "vacancy" in its atomic structure
and is unstable. Two different processes may occur to stabilise the ion
again.

The first process is fluorescence. Here one of the electrons from the outer
shells makes a radiative transition in order to occupy the vacancy. The
emitted photon has an energy corresponding to the energy difference
between the initial and final discrete states.

The other possibility to fill the gap is auto-ionisation through the
Auger process. In this case, also one of the electrons from the outer
shells fills the vacancy in the lower level. The released energy is not
emitted as a photon, however, but transferred to another electron from
the outer shells that is therefore able to escape from the ion. As a
result, the initial ionisation may lead to a double ionisation. If the
final electron configuration of the ion still has holes, more
auto-ionisations or fluorescence may follow until the ion has
stabilised.

In Fig. 4 the fluorescence yield
(probability that a vacancy will be filled by a radiative transition) is
shown for all elements. In general, the fluorescence yield increases
strongly with increasing nuclear charge Z, and is higher for the
innermost atomic shells. As a typical example, for Fe I a K-shell
vacancy has = 0.34,
while an Li-shell vacancy has
= 0.001. For O I these
numbers are 0.009 and 0, respectively.

Figure 4. Left panel:
Fluorescence yield
as a function of
atomic number Z for the K and L
shells. Right panel: Distribution of number of electrons liberated
after the initial removal of an electron from the K-shell (solid line)
or Li shell (dotted line), including the original
photo-electron, for Fe I; after
Kaastra &
Mewe (1993).

In Sect. 3.2.1 we showed how the collision of a
free electron with an ion can lead to ionisation. In principle, if the
free electron has insufficient energy (E < I), there will
be no ionisation. However, even in that case it is sometimes still
possible to ionise, but in a more complicated way. The process is as
follows. The collision can bring one of the electrons in the outermost
shells in a higher quantum level (excitation). In most cases, the ion
will return to its ground level by a radiative transition. But in some
cases the excited state is unstable, and a radiationless Auger
transition can occur (see Sect. 3.2.4). The vacancy
that is left behind by the
excited electron is being filled by another electron from one of the
outermost shells, while the excited electron is able to leave the ion
(or a similar process with the role of both electrons reversed).

Because of energy conservation, this process only occurs if the excited
electron comes from one of the inner shells (the ionisation potential
barrier must be taken anyhow). The process is in particular important
for Li-like and Na-like ions, and for several other individual atoms and
ions. As an example we treat here Li-like ions (see
Fig. 5, left panel). In that case the most
important contribution comes from a 1s-2p excitation.

Radiative recombination is the reverse process of photoionisation. A free
electron is captured by an ion while emitting a photon. The released
radiation is the so-called free-bound continuum emission. It is
relatively easy to show that there is a simple relation between the
photoionisation cross section
bf(E)
and the recombination cross
section fb,
namely the Milne-relation:

(13)

where gn is the statistical weight of the quantum
level into which the electron is captured (for an empty shell this is
gn = 2n2). By averaging over
a Maxwell distribution one gets the recombination-coefficient to level
n:

(14)

Of course there is energy conservation, so
E = 1/2 mev2 + I.

It can be shown that for the photoionisation cross section (9)
and for g = 1, constant and gn =
2n2:

(15)

With the asymptotic relations for the exponential integrals it can be
shown that

(16)(17)

Therefore for T
0 the recombination coefficient approaches
infinity: a cool plasma is hard to ionise. For T the
recombination coefficient goes to zero, because of the Milne relation
(v)
and because of the sharp decrease of the
photoionisation cross section for high energies.

As a rough approximation we can use further that I ~
(Z / n)2. Substituting
this we find that for kT << I (recombining plasmas)
Rn ~ n-1, while for kT
>> I (ionising plasmas) Rn ~
n-3. In
recombining plasmas in particular many higher excited levels will be
populated by the recombination, leading to significantly stronger line
emission. On the other hand, in ionising plasmas (such as supernova
remnants) recombination mainly occurs to the lowest levels. Note that
for recombination to the ground level the approximation (15) cannot be
used (the hydrogen limit), but instead one should use the exact
photoionisation cross section of the valence electron. By adding over
all values of n and applying an approximation
Seaton (1959)
found for the total radiative recombination rate
RR (in units
of m-3 s-1):

(18)

with EHZ2 /
kT and EH the Rydberg energy (13.6 eV). Note
that this equation only holds for hydrogen-like ions. For other ions
usually an analytical fit to numerical calculations is used:

(19)

where the approximation is strictly speaking only valid for T
near the equilibrium concentration. The approximations (16) and
(17) raise suspicion that for T 0 or T (19) could be a poor
choice.

The captured electron does not always reach the ground level immediately. We
have seen before that in particular for cool plasmas (kT <<
I) the higher excited levels are frequently populated. In order
to get to the ground
state, one or more radiative transitions are required. Apart from cascade
corrections from and to higher levels the recombination line radiation is
essentially given by (15). A comparison of recombination with
excitation tells that in particular for low temperatures (compared to
the line energy) recombination radiation dominates, and for high
temperatures excitation
radiation dominates. This is also one of the main differences between
photoionised and collisionally ionised plasmas, as photoionised plasmas in
general have a low temperature compared to the typical ionisation
potentials.

This process is more or less the inverse of excitation-autoionisation. Now a
free electron interacts with an ion, by which it is caught (quantum level
n" ") but
at the same time it excites an
electron from (n)
(n'
'). The doubly excited
state is in general not stable, and the ion will return to its original
state by
auto-ionisation. However there is also a possibility that one of the excited
electrons (usually the electron that originally belonged to the ion)
falls back
by a radiative transition to the ground level, creating therefore a stable,
albeit excited state (n"
") of the ion. In
particular excitations with '
= + 1 contribute much to this
process. In order to calculate this process, one should take account of many
combinations (n'
')(n"
").

The final transition probability is often approximated by

(20)

where A, B, T0 and T1
are adjustable parameters. Note that for T the
asymptotic behaviour is identical to the case of
radiative recombination. For T 0 however, dielectronic
recombination can be neglected; this is because the free electron has
insufficient energy to excite a bound electron. Dielectronic
recombination is a dominant process in the Solar corona, and also in
other situations it is often very important.

Dielectronic recombination produces more than one line photon. Consider for
example the dielectronic recombination of a He-like ion into a Li-like ion:

(21)

The first arrow corresponds to the electron capture, the second arrow to the
stabilising radiative transition 2p
1s and the third arrow
to the radiative transition 3s
2p of the captured
electron. This last
transition would have also occurred if the free electron was caught
directly into the 3s shell by normal radiative recombination. Finally,
the electron has
to decay further to the ground level and this can go through the normal
transitions in a Li-like ion (fourth arrow). This single recombination thus
produces three line photons.

Because of the presence of the extra electron in the higher orbit, the
energy h1 of
the 2p
1s transition is
slightly different from the energy in a normal He-like ion. The
stabilising transition is therefore also called a satellite
line. Because there are many different possibilities for the orbit of
the captured electron, one usually finds a forest of such satellite
lines surrounding the normal 2p
1s excitation line in
the He-like ion (or analogously for other iso-electronic sequences).
Fig. 6 gives an example of these satellite lines.

Figure 6. Spectrum of a plasma in
collisional ionisation equilibrium with kT = 2 keV, near the
Fe-K complex. Lines are labelled using the most
common designations in this field. The Fe XXV "triplet" consists of
the resonance line (w), intercombination line (actually split into x
and y) and the forbidden line (z). All other lines are satellite
lines. The labelled satellites are lines from Fe XXIV, most of the
lines with energy below the forbidden (z) line are from Fe XXIII. The
relative intensity of these satellites is a strong indicator for the
physical conditions in the source.

In most cases ionisation or recombination in collisionally ionised
plasmas is caused by interactions of an ion with a free electron. At low
temperatures (typically below 105 K) also charge transfer
reactions become important. During the interaction of two ions, an
electron may be transferred from one ion to the other; it is usually
captured in an excited state, and the excited ion is
stabilised by one or more radiative transitions. As hydrogen and helium
outnumber by at least a factor of 100 any other element (see
Table 3), in practice only
interactions between those elements
and heavier nuclei are important. Reactions with H I and He I lead
to recombination of the heavier ion, and reactions with H II and
He II to ionisation.

The electron captured during charge transfer recombination of an oxygen
ion (for instance O VII, O VIII) is usually captured in an intermediate
quantum state (principal quantum number n = 4-6). This leads to
enhanced line emission from that level as compared to the emission from
other principal levels, and this signature can be used to show the
presence of charge transfer reactions. Another signature - actually a
signature for all recombining plasmas - is of course the enhancement of
the forbidden line relative to the resonance
line in the O VII triplet (Sect. 5.2.2).

An important example is the charge transfer of highly charged ions from the
Solar wind with the neutral or weakly ionised Geocorona. Whenever the Sun is
more active, this process may produce enhanced thermal soft X-ray
emission in addition to the steady foreground emission from our own
Galaxy. See
Bhardwaj et
al. (2006)
for a review of X-rays from the Solar System. Care should
be taken not to confuse this temporary enhanced Geocoronal emission with
soft excess emission in a background astrophysical source.